# The LORE computational method in optimal control theory

Nedeljković, Nikola
(1985)
*The LORE computational method in optimal control theory.*
PhD thesis, Murdoch University.

## Abstract

A number of iterative algorithms for solving unconstrained continuous-time optimal control problems is developed. The method, named LORE is modelled locally on the linear quadratic problem and treats a whole family of algorithms in a unified manner, proposed approach is similar to the philosophy underlying Newton's, the Conjugate gradient and the quasi- Newton methods in the function minimization theory, The where the function being minimized is approximated locally by a quadratic function.

In addition to the development of a unified theory of algorithms, the thesis contains several first-order implementable algorithms, the convergence speed of which is comparable to that of second-order methods.

A proof of the reduction of the cost at each iterative step of the LORE algorithms, a convergence analysis in the Lm∞ space and a proof of the convergence in the space of relaxed controls are included.

The power of the adopted approach lies in the use of the Riccati matrix differential equation which within the context of the LQRE method always has a bounded solution. Within the general framework of the analysis it is possible to obtain both first-order and second-order algorithms. The emphasis is however placed on the first-order LORE algorithms which are simpler and computationally less demanding per iterative step; their computational effectiveness is compared with the performance of known methods.

There is a noticeable degree of similarity in the form of the differential equations used by the LORE algorithms and the differential equations in the well-known second order methods; in fact, it is possible to derive LORE variants of the second variation and differential dynamic programming methods.

The LORE algorithms converge in one step on the linear quadratic problem and are well suited for solving nonlinear problems with linear constraints via the penalty Their application in the computation of the singular optimal control, by adding and subtracting a quadratic term to the cost, is suggested. function methods.

The method has been extended to handle problems with terminal equality constraints, control constraints (LORE projection technique) and a class of state and control equality constraints (sequential LORE-restoration algorithm). The LORE method of discrete-time unconstrained systems has also been developed.

Item Type: | Thesis (PhD) |
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Murdoch Affiliation: | School of Mathematical and Physical Sciences |

Supervisor(s): | Kloeden, Peter E. |

URI: | http://researchrepository.murdoch.edu.au/id/eprint/51530 |

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